A052344 -id:A052344 - cp's OEIS frontend

A053587 Indices of A052344 (ways to write n as sum of two nonzero triangular numbers) where record values are reached.

2, 16, 81, 471, 1056, 1381, 6906, 17956, 34531, 40056, 200281, 520731, 1001406, 1482081, 7410406, 19267056, 37052031, 60765331, 303826656, 789949306, 1519133281, 3220562556, 13429138206, 16102812781, 41867313231, 80514063906, 196454315931, 711744324931

Offset: 1

Jeremy Rouse, Jan 19 2000

The subsequence of primes begins: 2, 1381, 1519133281 [Jonathan Vos Post, Feb 01 2011].

			The order of the terms is ignored when deciding in how many ways the sum can be expressed. For example, a(2) does not equal 9, although 9 = 3 + 6 = 6 + 3.
a(2) = 16 because 16 = 1 + 15 = 6 + 10. a(3) = 81 because 81 = 3 + 78 = 15 + 66 = 36 + 55.

Kenneth Korbin, Problem 665, The College Mathematics Journal Vol. 30 No. 5, Nov. 1999. Asks for the 48th number in this sequence.

Probably differs from A052348 only at n=1, 2, 4.

Cf. A000217, A052343, A052344, A052345, A052346, A052347, A052348.

More terms from Christian G. Bower, Jan 23 2000
a(25)-a(26) from Donovan Johnson, Jun 26 2010
a(27)-a(28) from Donovan Johnson, Mar 20 2013

A052347 Record values reached in A052343 and A052344.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 18, 20, 24, 32, 36, 40, 48, 64, 72, 80, 96, 108, 128, 144, 160, 192, 216

Offset: 1

Christian G. Bower, Jan 23 2000

Cf. A052343-A052348, A053587.

a(25)-a(26) from Donovan Johnson, Jun 26 2010

a(27)-a(28) from Donovan Johnson, Mar 20 2013

A052343 Number of ways to write n as the unordered sum of two triangular numbers (zero allowed).

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 2, 1, 0, 1, 1, 1, 1, 1, 0, 1, 2, 0, 1, 0, 1, 2, 1, 0, 1, 1, 0, 1, 1, 1, 1, 2, 0, 0, 1, 0, 2, 1, 1, 1, 0, 0, 2, 1, 0, 1, 2, 0, 1, 1, 0, 2, 0, 0, 0, 2, 2, 1, 1, 0, 1, 1, 0, 0, 1, 1, 2, 1, 0, 1, 1, 0, 2, 1, 0, 0, 2, 0, 1, 1, 0, 3, 0, 1, 1, 0, 0, 1, 1, 0, 1, 2, 1, 1, 2, 0, 0, 1, 0, 1, 1, 1

Offset: 0

Christian G. Bower, Jan 23 2000

Number of ways of writing n as a sum of a square and twice a triangular number (zeros allowed). - Michael Somos, Aug 18 2003
a(A020757(n))=0; a(A020756(n))>0; a(A119345(n))=1; a(A118139(n))>1. - Reinhard Zumkeller, May 15 2006
Also, number of ways to write 4n+1 as the unordered sum of two squares of nonnegative integers. - Vladimir Shevelev, Jan 21 2009
The average value of a(n) for n <= x is Pi/4 + O(1/sqrt(x)). - Vladimir Shevelev, Feb 06 2009

			G.f. = 1 + x + x^2 + x^3 + x^4 + 2*x^6 + x^7 + x^9 + x^10 + x^11 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Vladimir Shevelev, Binary additive problems: recursions for numbers of representations, arXiv:0901.3102 [math.NT], 2009-2013.
Vladimir Shevelev, Binary additive problems: theorems of Landau and Hardy-Littlewood type, arXiv:0902.1046 [math.NT], 2009-2012.

Cf. A000217, A052344, A052345 (greedy inverse), A052346, A052347, A052348, A053587, A056303, A056304.
Cf. A053692, A093518, A121444, A259285, A259287, A260415, A260516.
Cf. A005369, A010052.

a052343 = (flip div 2) . (+ 1) . a008441
-- Reinhard Zumkeller, Jul 25 2014

A052343 := proc(n)
    local a,t1idx,t2idx,t1,t2;
    a := 0 ;
    for t1idx from 0 do
        t1 := A000217(t1idx) ;
        if t1 > n then
            break;
        end if;
        for t2idx from t1idx do
            t2 := A000217(t2idx) ;
            if t1+t2 > n then
                break;
            elif t1+t2 = n then
                a := a+1 ;
            end if;
        end do:
    end do:
    a ;
end proc: # R. J. Mathar, Apr 28 2020

Length[PowersRepresentations[4 # + 1, 2, 2]] & /@ Range[0, 101] (* Ant King, Dec 01 2010 *)
d1[k_]:=Length[Select[Divisors[k],Mod[#,4]==1&]];d3[k_]:=Length[Select[Divisors[k],Mod[#,4]==3&]];f[k_]:=d1[k]-d3[k];g[k_]:=If[IntegerQ[Sqrt[4k+1]],1/2 (f[4k+1]+1),1/2 f[4k+1]];g[#]&/@Range[0,101] (* Ant King, Dec 01 2010 *)
a[ n_] := Length @ Select[ Table[ Sqrt[n - i - i^2], {i, 0, Quotient[ Sqrt[4 n + 1] - 1, 2]}], IntegerQ]; (* Michael Somos, Jul 28 2015 *)
a[ n_] := Length @ FindInstance[ {j >= 0, k >= 0, j^2 + k^2 + k == n}, {k, j}, Integers, 10^9]; (* Michael Somos, Jul 28 2015 *)

{a(n) = if( n<0, 0, sum(i=0, (sqrtint(4*n + 1) - 1)\2, issquare(n - i - i^2)))}; /* Michael Somos, Aug 18 2003 */

a(n) = ceiling(A008441(n)/2). - Reinhard Zumkeller, Nov 03 2009
G.f.: (Sum_{k>=0} x^(k^2 + k)) * (Sum_{k>=0} x^(k^2)). - Michael Somos, Aug 18 2003
Recurrence: a(n) = Sum_{k=1..r(n)} r(2n-k^2+k) - C(r(n),2) - a(n-1) - a(n-2) - ... - a(0), n>=1,a (0)=1, where r(n)=A000194(n+1) is the nearest integer to square root of n+1. For example, since r(6)=3, a(6) = r(12) + r(10) + r(6) - C(3,2) - a(5) - ... - a(0) = 4 + 3 + 3 - 3 - 0 - 1 - 1 - 1 - 1 - 1 = 2. - Vladimir Shevelev, Feb 06 2009
a(n) = A025426(8n+2). - Max Alekseyev, Mar 09 2009
a(n) = (A002654(4n+1) + A010052(4n+1)) / 2. - Ant King, Dec 01 2010
a(2*n + 1) = A053692(n). a(4*n + 1) = A259287(n). a(4*n + 3) = A259285(n). a(6*n + 1) = A260415(n). a(6*n + 4) = A260516(n). - Michael Somos, Jul 28 2015
a(3*n) = A093518(n). a(3*n + 1) = A121444(n). a(9*n + 2) = a(n). a(9*n + 5) = a(9*n + 8) = 0. - Michael Somos, Jul 28 2015
Convolution of A005369 and A010052. - Michael Somos, Jul 28 2015

A319797 Number T(n,k) of partitions of n into exactly k positive triangular numbers; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 2, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 2, 1, 1, 1, 1, 1, 0, 1, 0, 1

Offset: 0

Alois P. Heinz, Sep 28 2018

Equals A181506 when the first column is removed. - Georg Fischer, Jul 26 2023

			Triangle T(n,k) begins:
  1;
  0, 1;
  0, 0, 1;
  0, 1, 0, 1;
  0, 0, 1, 0, 1;
  0, 0, 0, 1, 0, 1;
  0, 1, 1, 0, 1, 0, 1;
  0, 0, 1, 1, 0, 1, 0, 1;
  0, 0, 0, 1, 1, 0, 1, 0, 1;
  0, 0, 1, 1, 1, 1, 0, 1, 0, 1;
  0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1;
  0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1;
  0, 0, 1, 2, 1, 1, 1, 1, 1, 0, 1, 0, 1;

Alois P. Heinz, Rows n = 0..200, flattened

Columns k=0-10 give: A000007, A010054 (for n>0), A052344, A063993, A319814, A319815, A319816, A319817, A319818, A319819, A319820.
Row sums give A007294.
T(2n,n) gives A319799.
Cf. A000217, A117278, A181506, A243148, A319394.

h:= proc(n) option remember; `if`(n<1, 0,
      `if`(issqr(8*n+1), n, h(n-1)))
    end:
b:= proc(n, i) option remember; `if`(n=0 or i=1, x^n,
      b(n, h(i-1))+expand(x*b(n-i, h(min(n-i, i)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, h(n))):
seq(T(n), n=0..20);

h[n_] := h[n] = If[n < 1, 0, If[IntegerQ @ Sqrt[8*n + 1], n, h[n - 1]]];
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x^n, b[n, h[i - 1]] + Expand[ x*b[n - i, h[Min[n - i, i]]]]];
T[n_] := Table[Coefficient[#, x, i], {i, 0, n}]& @ b[n, h[n]];
Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, May 27 2019, after Alois P. Heinz *)

T(n,k) = [x^n y^k] 1/Product_{j>=1} (1-y*x^A000217(j)).

A307597 Number of partitions of n into 2 distinct positive triangular numbers.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 2, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 2, 0, 1, 1, 0, 2, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 2, 0, 0, 1, 0, 3, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 2, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 3, 0, 1

Offset: 0

Ilya Gutkovskiy, Apr 17 2019

The greedy inverse (positions of first occurrence of n) starts 0, 4, 16, 81, 471, 2031, 1381, 11781, 6906, 17956, ... - R. J. Mathar, Apr 28 2020

			a(16) = 2 because we have [15, 1] and [10, 6].

Alois P. Heinz, Table of n, a(n) for n = 0..65536 (first 10000 terms from David A. Corneth)

Cf. A000217, A008441, A024940, A025441, A052344, A053603, A260647, A307598.

Cf. A010054.

a(n) = [x^n y^2] Product_{k>=1} (1 + y*x^(k*(k+1)/2)).

a(n) = Sum_{k=1..floor((n-1)/2)} c(k) * c(n-k), where c = A010054. - Wesley Ivan Hurt, Jan 06 2024

A230121 Number of ways to write n = x + y + z (0 < x <= y <= z) such that x(x+1)/2 + y(y+1)/2 + z*(z+1)/2 is a triangular number.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 1, 2, 1, 1, 0, 2, 1, 2, 1, 2, 3, 2, 2, 6, 1, 3, 5, 1, 2, 3, 5, 2, 1, 3, 3, 3, 4, 3, 8, 2, 5, 11, 2, 5, 8, 4, 6, 4, 9, 4, 6, 5, 4, 6, 3, 8, 8, 5, 8, 10, 7, 7, 11, 8, 6, 7, 8, 5, 9, 7, 6, 8, 7, 7, 8, 13, 9, 11, 10, 7, 22, 9, 10, 13, 3, 6, 10, 8, 17, 12, 7, 9, 10, 16, 6, 18, 18, 10, 15, 9, 12, 20, 5

Offset: 1

Zhi-Wei Sun, Oct 10 2013

nonn

Conjecture: (i) a(n) > 0 except for n = 1, 2, 4, 5, 7, 12. Moreover, for each n = 20, 21, ... there are three distinct positive integers x, y and z with x + y + z = n such that x*(x+1)/2 + y*(y+1)/2 + z*(z+1)/2 is a triangular number.
(ii) A positive integer n cannot be written as x + y + z (x, y, z > 0) with x^2 + y^2 + z^2 a square if and only if n has the form 2^r*3^s or the form 2^r*7, where r and s are nonnegative integers.
(iii) Any integer n > 14 can be written as a + b + c + d, where a, b, c, d are positive integers with a^2 + b^2 + c^2 + d^2 a square. If n > 20 is not among 22, 28, 30, 38, 44, 60, then we may require additionally that a, b, c, d are pairwise distinct.
(iv) For each integer n > 50 not equal to 71, there are positive integers a, b, c, d with a + b + c + d = n such that both a^2 + b^2 and c^2 + d^2 are squares.
Part (ii) and the first assertion in part (iii) were confirmed by Chao Huang and Zhi-Wei Sun in 2021. - Zhi-Wei Sun, May 09 2021

			a(16) = 1 since 16 = 3 + 6 + 7 and 3*4/2 + 6*7/2 + 7*8/2 = 55 = 10*11/2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..3000
Chao Huang and Zhi-Wei Sun, On partitions of integers with restrictions involving squares, arXiv:2105.03416 [math.NT], 2021.
Zhi-Wei Sun, Diophantine problems involving triangular numbers and squares, a message to Number Theory List, Oct. 11, 2013.

Cf. A000217, A000290.

Cf. A008443, A053604, A053603, A052343, A052344, A008441.

SQ[n_]:=IntegerQ[Sqrt[n]]
T[n_]:=n(n+1)/2
a[n_]:=Sum[If[SQ[8(T[i]+T[j]+T[n-i-j])+1],1,0],{i,1,n/3},{j,i,(n-i)/2}]
Table[a[n],{n,1,100}]

a(n)=my(t=(n+1)*n/2,s);sum(x=1,n\3,s=t-n--*x;sum(y=x,n\2,is_A000217(s-(n-y)*y))) \\ - M. F. Hasler, Oct 11 2013

A052346 Smallest number which is the sum of two positive triangular numbers in exactly n different ways.

Original entry on oeis.org

1, 2, 16, 81, 471, 1056, 1381, 11781, 6906, 17956, 34531, 123256, 40056, 4462656, 305256, 448906, 200281, 1957231, 520731, 10563906, 1001406, 11222656, 539550781, 3454506, 1482081, 75865156, 7172606106, 8852431, 25035156, 334020781, 13018281, 38531031, 7410406, 7014160156

Offset: 0

Christian G. Bower, Jan 23 2000

nonn

From Chai Wah Wu, Oct 20 2023: (Start)
Other terms:
a(35) =  42980356
a(36) =  19267056
a(38) =  1289707656
a(39) =  2782318906
a(40) =  37052031
a(41) =  256720506
a(42) =  325457031
a(45) =  221310781
a(47) =  550240551
a(48) =  60765331
a(50) =  2200089531
a(54) =  327539956
a(56) =  926300781
a(59) =  7629645156
a(60) =  481676406
a(63) =  4598740656
a(64) =  303826656
a(68) =  6418012656
a(71) =  4579579956
a(72) =  789949306
a(80) =  1519133281
a(81) =  9498658731
a(84) =  12041910156
a(90) =  8188498906
a(96) =  3220562556
a(108) =  13429138206
(End)

			a(4) = 471 because 471 is the sum of two positive triangular numbers in exactly 4 different ways (as 300+171, 351+120, 435+36, and 465 + 6), and there is no smaller number that has this property.

Cf. A000217, A052343, A052344, A052345, A052347, A052348, A053587, A064816.

a(27), a(28) = 8852431, 25035156; a(26) not yet found
a(26) from Donovan Johnson, Nov 17 2008
Name edited (added the qualifier "positive"), example edited, and a(29)-a(32) added by Jon E. Schoenfield, Jul 16 2017
a(33) from Chai Wah Wu, Oct 20 2023

A052348 Indices of A052343 (ways to write n as sum of two triangular numbers) where record values are reached.

Original entry on oeis.org

0, 6, 81, 276, 1056, 1381, 6906, 17956, 34531, 40056, 200281, 520731, 1001406, 1482081, 7410406, 19267056, 37052031, 60765331, 303826656, 789949306, 1519133281, 3220562556, 13429138206, 16102812781, 41867313231, 80514063906, 196454315931, 711744324931

Offset: 1

Christian G. Bower, Jan 23 2000

nonn

			a(2) = 6 because 6 = 6 + 0 = 3 + 3. a(3) = 81 because 81 = 3 + 78 = 15 + 66 = 36 + 55.

Kenneth Korbin, Problem 665, The College Mathematics Journal Vol. 30 No. 5, Nov. 1999.

Probably differs from A053587 only at n=1, 2, 4.

Cf. A000217, A052343, A052344, A052345, A052346, A052347.

a(23)-a(26) from Donovan Johnson, May 24 2009

a(27)-a(28) from Donovan Johnson, Mar 20 2013

A052345 Least k such that A052343(k)=n.

Original entry on oeis.org

5, 1, 6, 81, 276, 1056, 1381, 50781, 6906, 17956, 34531, 660156, 40056, 4462656, 305256, 448906, 200281, 412597656, 520731, 12397766113281, 1001406, 11222656, 539550781, 7631406, 1482081, 75865156, 422394133, 8852431, 25035156, 161170959472656

Offset: 0

Christian G. Bower, Jan 23 2000

nonn

Kenneth Korbin, Problem 665, The College Mathematics Journal Vol. 30 No. 5, Nov. 1999.

Cf. A052343, A052344, A052346, A052347, A052348, A053587.

a(19) and a(29) from Max Alekseyev, Mar 09 2009

a(19) corrected by Max Alekseyev, Mar 11 2009

A185979 Numbers which are the sum of two positive triangular numbers in more than one way.

Original entry on oeis.org

16, 31, 42, 46, 51, 56, 72, 76, 81, 94, 106, 111, 121, 123, 126, 133, 141, 146, 156, 157, 172, 174, 181, 186, 191, 196, 198, 211, 216, 225, 226, 231, 237, 241, 246, 256, 259, 268, 276, 281, 286, 289, 291, 297, 301, 306, 310, 315, 321, 326, 328, 331, 336, 342, 346, 354, 361, 366, 367

Offset: 1

Wolfdieter Lang, Feb 15 2011

This is a subsequence of A020756 (sums of two triangular numbers).
This is also a subsequence of A051533 (sums of two positive triangular numbers). This is not a subsequence of A185978 (nontriangular numbers as sums of (positive) triangular numbers). E.g., a(32)=231 is missing there because 231=A000217(21). See A185980.
For the numbers which are sums of two positive triangular numbers in exactly two ways see A064816.
The first number which can be written in exactly three ways as sums of positive triangular numbers is 81.
a(n) gives the positions where A052344 entries are >= 2: A052344(a(n)) >= 2.

			16 = 15 + 1 = 10 + 6.
81 = 45 + 36 = 66 + 15 = 78 + 3.
231= 210 + 21 = 153 + 78

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000217, A020756, A051533, A052344, A064816, A185980.

cp's OEIS Frontend

A053587 Indices of A052344 (ways to write n as sum of two nonzero triangular numbers) where record values are reached.

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A052347 Record values reached in A052343 and A052344.

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A052343 Number of ways to write n as the unordered sum of two triangular numbers (zero allowed).

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A319797 Number T(n,k) of partitions of n into exactly k positive triangular numbers; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A307597 Number of partitions of n into 2 distinct positive triangular numbers.

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A230121 Number of ways to write n = x + y + z (0 < x <= y <= z) such that x*(x+1)/2 + y*(y+1)/2 + z*(z+1)/2 is a triangular number.

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A052346 Smallest number which is the sum of two positive triangular numbers in exactly n different ways.

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A052348 Indices of A052343 (ways to write n as sum of two triangular numbers) where record values are reached.

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A052345 Least k such that A052343(k)=n.

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A230121 Number of ways to write n = x + y + z (0 < x <= y <= z) such that x(x+1)/2 + y(y+1)/2 + z*(z+1)/2 is a triangular number.